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Consider a scenario similar to metal rod with temperature varying across the length. o Like a metal spoon dipped in a hot curry and removed o Or a piece of iron heated in a furnace and then removed And we want to study the heat flow, how the temperature at different points vary with time P Consider the metal rod to be 1 D Neglect o Heat loss to environment o Any heating source The 1 D rod is thus isolated X Consider the metal rod to be a cylindrical rod oOf radius R and length L It is given that the rod is not in contact with any heat source We assume that no heat is lost to surroundings V The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 L The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] 𝑑𝑇(𝑥,𝑡)𝑑𝑡 𝑄(𝑥,𝑡) Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 =𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] + 𝑘𝑄(𝑥,𝑡) where c, k’ are constants. where Q (x,t) is the heat absorbed by the element at x T From calculus we have [𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑘𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) where 𝑘 =𝑐(𝑑𝑥2) which describes how T at some x varies with t. G The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 R Place the rod in a mathematical grid Let T (x, t) denote the temperature of any mass point x at time t. Z The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 = 𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] where c is a constant. M From calculus we have [𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 = 𝑘𝜕2𝑇𝜕𝑥2 which describes how T at some x varies with t. Here 𝑘 =𝑐(𝑑𝑥2). D BLANK Consider a scenario similar to metal rod with temperature varying across the length. o Like a metal spoon dipped in a hot curry and removed o Or a piece of iron heated in a furnace and then removed And we want to study the heat flow, how the temperature at different points vary with time P Consider the metal rod to be 1 D Neglect o Heat loss to environment o Any heating source The 1 D rod is thus isolated X Consider the metal rod to be a cylindrical rod oOf radius R and length L It is given that the rod is not in contact with any heat source We assume that no heat is lost to surroundings V The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 L The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] 𝑑𝑇(𝑥,𝑡)𝑑𝑡 𝑄(𝑥,𝑡) Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 =𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] + 𝑘𝑄(𝑥,𝑡) where c, k’ are constants. where Q (x,t) is the heat absorbed by the element at x T From calculus we have [𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑘𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) where 𝑘 =𝑐(𝑑𝑥2) which describes how T at some x varies with t. G The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 R Place the rod in a mathematical grid Let T (x, t) denote the temperature of any mass point x at time t. Z The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 = 𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] where c is a constant. M From calculus we have [𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 = 𝑘𝜕2𝑇𝜕𝑥2 which describes how T at some x varies with t. Here 𝑘 =𝑐(𝑑𝑥2). D BLANK Consider a scenario similar to metal rod with temperature varying across the length. o Like a metal spoon dipped in a hot curry and removed o Or a piece of iron heated in a furnace and then removed And we want to study the heat flow, how the temperature at different points vary with time P Consider the metal rod to be 1 D Neglect o Heat loss to environment o Any heating source The 1 D rod is thus isolated X Consider the metal rod to be a cylindrical rod oOf radius R and length L It is given that the rod is not in contact with any heat source We assume that no heat is lost to surroundings V The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 L The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] 𝑑𝑇(𝑥,𝑡)𝑑𝑡 𝑄(𝑥,𝑡) Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 =𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] + 𝑘𝑄(𝑥,𝑡) where c, k’ are constants. where Q (x,t) is the heat absorbed by the element at x T From calculus we have [𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑘𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) where 𝑘 =𝑐(𝑑𝑥2) which describes how T at some x varies with t. G The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 R Place the rod in a mathematical grid Let T (x, t) denote the temperature of any mass point x at time t. Z The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 = 𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] where c is a constant. M From calculus we have [𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 = 𝑘𝜕2𝑇𝜕𝑥2 which describes how T at some x varies with t. Here 𝑘 =𝑐(𝑑𝑥2). D BLANK Consider a scenario similar to metal rod with temperature varying across the length. o Like a metal spoon dipped in a hot curry and removed o Or a piece of iron heated in a furnace and then removed And we want to study the heat flow, how the temperature at different points vary with time P Consider the metal rod to be 1 D Neglect o Heat loss to environment o Any heating source The 1 D rod is thus isolated X Consider the metal rod to be a cylindrical rod oOf radius R and length L It is given that the rod is not in contact with any heat source We assume that no heat is lost to surroundings V The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 L The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] 𝑑𝑇(𝑥,𝑡)𝑑𝑡 𝑄(𝑥,𝑡) Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 =𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] + 𝑘𝑄(𝑥,𝑡) where c, k’ are constants. where Q (x,t) is the heat absorbed by the element at x T From calculus we have [𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑘𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) where 𝑘 =𝑐(𝑑𝑥2) which describes how T at some x varies with t. G The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 R Place the rod in a mathematical grid Let T (x, t) denote the temperature of any mass point x at time t. Z The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 = 𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] where c is a constant. M From calculus we have [𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 = 𝑘𝜕2𝑇𝜕𝑥2 which describes how T at some x varies with t. Here 𝑘 =𝑐(𝑑𝑥2). D BLANK Consider a scenario similar to metal rod with temperature varying across the length. o Like a metal spoon dipped in a hot curry and removed o Or a piece of iron heated in a furnace and then removed And we want to study the heat flow, how the temperature at different points vary with time P Consider the metal rod to be 1 D Neglect o Heat loss to environment o Any heating source The 1 D rod is thus isolated X Consider the metal rod to be a cylindrical rod oOf radius R and length L It is given that the rod is not in contact with any heat source We assume that no heat is lost to surroundings V The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 L The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] 𝑑𝑇(𝑥,𝑡)𝑑𝑡 𝑄(𝑥,𝑡) Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 =𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] + 𝑘𝑄(𝑥,𝑡) where c, k’ are constants. where Q (x,t) is the heat absorbed by the element at x T From calculus we have [𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑘𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) where 𝑘 =𝑐(𝑑𝑥2) which describes how T at some x varies with t. G The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 R Place the rod in a mathematical grid Let T (x, t) denote the temperature of any mass point x at time t. Z From calculus we have [𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 = 𝑘𝜕2𝑇𝜕𝑥2 which describes how T at some x varies with t. Here 𝑘 =𝑐(𝑑𝑥2). D BLANK The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 = 𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] where c is a constant. M The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 = 𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] where c is a constant. M Consider a scenario similar to metal rod with temperature varying across the length. o Like a metal spoon dipped in a hot curry and removed o Or a piece of iron heated in a furnace and then removed And we want to study the heat flow, how the temperature at different points vary with time P Consider the metal rod to be 1 D Neglect o Heat loss to environment o Any heating source The 1 D rod is thus isolated X Consider the metal rod to be a cylindrical rod oOf radius R and length L It is given that the rod is not in contact with any heat source We assume that no heat is lost to surroundings V The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 L The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] 𝑑𝑇(𝑥,𝑡)𝑑𝑡 𝑄(𝑥,𝑡) Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 =𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] + 𝑘𝑄(𝑥,𝑡) where c, k’ are constants. where Q (x,t) is the heat absorbed by the element at x T From calculus we have [𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑘𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) where 𝑘 =𝑐(𝑑𝑥2) which describes how T at some x varies with t. G The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 R Place the rod in a mathematical grid Let T (x, t) denote the temperature of any mass point x at time t. Z From calculus we have [𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 = 𝑘𝜕2𝑇𝜕𝑥2 which describes how T at some x varies with t. Here 𝑘 =𝑐(𝑑𝑥2). D BLANK From calculus we have [𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 = 𝑘𝜕2𝑇𝜕𝑥2 which describes how T at some x varies with t. Here 𝑘 =𝑐(𝑑𝑥2). D The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 = 𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] where c is a constant. M Consider a scenario similar to metal rod with temperature varying across the length. o Like a metal spoon dipped in a hot curry and removed o Or a piece of iron heated in a furnace and then removed And we want to study the heat flow, how the temperature at different points vary with time P Consider the metal rod to be 1 D Neglect o Heat loss to environment o Any heating source The 1 D rod is thus isolated X Consider the metal rod to be a cylindrical rod oOf radius R and length L It is given that the rod is not in contact with any heat source We assume that no heat is lost to surroundings V The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 L The time rate of change of temperature of a point x is proportional to the difference in temperature with the neighbouring points. 𝑑𝑇(𝑥,𝑡)𝑑𝑡 [𝑇(𝑥 + 𝑑𝑥) 𝑇(𝑥)] + [𝑇(𝑥 𝑑𝑥) 𝑇(𝑥)] 𝑑𝑇(𝑥,𝑡)𝑑𝑡 𝑄(𝑥,𝑡) Implies, 𝑑𝑇(𝑥,𝑡) 𝑑𝑡 =𝑐[𝑇(𝑥 + 𝑑𝑥) 2𝑇(𝑥) + 𝑇(𝑥 𝑑𝑥)] + 𝑘𝑄(𝑥,𝑡) where c, k’ are constants. where Q (x,t) is the heat absorbed by the element at x T From calculus we have [𝑇(𝑥 + 𝑑𝑥) − 2𝑇(𝑥) + 𝑇(𝑥 − 𝑑𝑥)] =(𝑑𝑥2)𝑑2𝑇𝑑𝑥2 Combining both equations we get, 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑐(𝑑𝑥2)𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) or 𝜕𝑇(𝑥,𝑡)𝜕𝑡 =𝑘𝜕2𝑇𝜕𝑥2 + 𝑘𝑄(𝑥,𝑡) where 𝑘 =𝑐(𝑑𝑥2) which describes how T at some x varies with t. G The temperature (T) of the rod is different at different points and vary with time To study how T evolves, consider the rod to be made of mass points separated by dx We consider the limit dx 0 R Place the rod in a mathematical grid Let T (x, t) denote the temperature of any mass point x at time t. Z BLANK



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